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Math Help - Poisson Distribution Problem

  1. #1
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    Poisson Distribution Problem

    Assume that when the German plays Hungary in soccer, each team scores independently as a homogeneous Poisson process with rates λGermany = 1 and λHungary = 3 goals per game.

    a) Expected number of total goals in a single game.

    b) Expected number of total goals scored in the first half, given that 2 total goals were scored in the entire game.

    c) Expected number of goals Germany scored in the entire game, given that 2 total goals were scored in the first half.


    What I really need help with is getting the formulas for parts b & c. I'm having difficulty forming what the Expectation formula should be. `
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  2. #2
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    Re: Poisson Distribution Problem

    Hey Rorshach27.

    Hint: If two process are independent then P(A = a, B = b) = P(A = a) * P(B = b) and P(A|B) = P(A = a,B = b)/P(B=b)
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  3. #3
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    Re: Poisson Distribution Problem

    Thanks for the help,

    at this point I'm still having trouble with part c of this problem.

    I've come to the conclusion that if I use the conditional expectation of


    E(X|X + Y = n) = (λ1 / λ1 + λ2) * n
    I might get to the answer, however I'm not sure how to account for the fact that our given statement states two goals were scored in the first half

    Any help?
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  4. #4
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    Re: Poisson Distribution Problem

    I think you have to redefine the rates on a per half rather than per game basis. I.e

    $$\lambda_{Germany}=0.5 \mbox{ and }\lambda_{Hungary}=1.5$$
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